International Parity Conditions and Market Risk Thomas C. Chiang, Ph.D. Austin Professor of Finance Drexel University Date: March 15, 2003 Keywords: International Asset Pricing , Purchasing Power Parity, Uncovered Interest Parity, Exchange Rate Risk, Equity Premiums, Real Interest Rate Parity Correspondence Thomas C. Chiang Department of Finance, Drexel University 3141 Chestnut Street, Philadelphia, Pa. 19104, USA Tel: (215) 895-1279; Fax: (215) 895-2955 Email: [email protected]File: International Parity Conditions_v2
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International Parity Conditions and Market Risk
Thomas C. Chiang, Ph.D. Austin Professor of Finance
Drexel University Date: March 15, 2003 Keywords: International Asset Pricing , Purchasing Power Parity, Uncovered Interest Parity, Exchange Rate Risk, Equity Premiums, Real Interest Rate Parity Correspondence Thomas C. Chiang Department of Finance, Drexel University 3141 Chestnut Street, Philadelphia, Pa. 19104, USA Tel: (215) 895-1279; Fax: (215) 895-2955 Email: [email protected] File: International Parity Conditions_v2
Notes: a. An asterisk indicates statistically significant difference from zero at the 1% level. b. The numbers in parentheses are absolute values of the t-statistics. c. The joint test is to test: ( 0β 1β )´ = (0 1)´
11
parity in Panel A are negligible and statistically insignificant. These results, together with the
low R-squares of the test equation, render no supportive evidence on the parallel movements of
the two exchange-rate adjusted interest rates. However, this also underscores the independence
of interest-rate policies during the floating exchange rate period.
The results for estimating the international stock-return parity are presented in Panel B. The
estimated coefficients indicate that correlations with the US market are positive and statistically
significant.8 The values of the coefficients vary within a very narrow range, from 0.625 to 0.685
across the different European markets, supporting the efficient aspect of comovements of
international stock returns. However, the test results are still unable to provide supporting
evidence for the null hypothesis ( 0β 1β )´ = (0 1)´. The rejection of the null implies the
violation of the international stock-return parity. This is understandable since, in addition to
preference differences and possible asymmetrical information (Frankel and Schmukler, 2000),
the index composition varies among the nations, and the underlying industries are subject to their
inherent, different volatility and price/ interest rate sensibility (Roll, 1992).
Panel C reports estimates of the purchasing-power parity relative to the United States.
Again, the estimated slopes are far from unity. None of the R-squares exceeds the 2% level.
This result is very comparable to those reported by Krugman (1978), Roll (1979), Frenkel
(1981), Solnik (1982), and Mishkin (1984), among others.9 This result is not surprising since our
data sample is relatively short, while most of the evidence in favor of the PPP employs much
longer data spans. For example, Abuaf and Jorion (1990), Lothian and Taylor (1997b), Jorion
and Sweeney (1996), Cheung and Lai (1993, 1998), Fleissig and Strauss (2000), and Baum et al
(2001) are able to find evidence of mean-reversion in deviations from purchasing power parity.
The failure of PPP in the short run is perceivable since, in the very nature of price behavior,
commodity prices are relatively sticky and exchange rates behave more or less like asset prices.
Thus, the change in exchange rates as they adjust to news appears to be more sensitive and
effective than that of the commodity prices. In addition, failure to achieve PPP in the short run
12
may also result from differing compositions in consumer-price indices across different countries
(Patel, 1990), differing productivity shocks (Fisher and Park, 1991), and measurement errors in
prices results from aggregation (Talor, 1988; Cheung and Lai, 1993). 10
4. Deviations from Parity Conditions and Risk
4.1. Source of Deviations
The analysis in Section 2 conveys two important messages: international parity conditions
are interrelated; departure from parity conditions is commonly associated with real interest-rate
differentials and equity-premium differentials. Although some earlier researchers (Korajczyk,
1985; Levine, 1989; Huang, 1990; Chiang, 1991; Korajczyk and Viallet, 1992) recognize these
key elements, their studies merely focus on a single parity (Hodrick, 1987) or a smaller set of
parity conditions (Mishkin, 1984; Marston, 1997); an explicit role of international stock market
was excluded from their analysis. The current study extends previous research by incorporating
the linkage of stock markets into an integrated financial system. This research is bound to
provide more insight into the multi-market analysis of international asset allocation, offering a
broader spectrum of portfolio behavior in a general equilibrium framework.
To illustrate, assume that expected changes in spot exchange rates can be predicted by a
linear relation of the expected inflation-rate differential, short-term interest-rate differential, and
expected national-stock-return differential as implied by the three parity conditions. Thus, we
write:
(13) )()()( *11
**111
et
ettt
et
et
et RRrrpps +++++ −+−+∆−∆=∆ γρα
The arguments on the right side of Equation (13) are considered to be the key variables that
affect international transactions involving a nation’s balance of payments. In particular, the
variable of the expected inflation differential affects trade flows in a country’s current account,11
while the other two arguments govern capital flows involving bonds and stocks in the capital
accounts. The weight of each component will be reflected, respectively, in the parametersα, ρ ,
and γ , and the restriction α + ρ + γ = 1 is constrained by the sum of components of the balance
13
of payments. Subtracting ( from both sides of the equation (13) and rearranging the
variables yield:
)*tt rr −
[( 1 tet rR −+
12
( *1 t
et rR + −
[()]*tr −−
()t Rr −−
(14) )]()[()]())( **11
**1
*1 t
ett
ett
ettt
et rprprRrrs −∆−−∆+−−=−−∆ ++++ αγ
An important message emerging from Equation (14) is that the deviation from UIP is essentially
due to the excess relative returns prevailing in stock and goods markets as compared with the
risk-free rate in the bond markets. In the earlier study by Giovannini and Jorion (1987), they
found evidence that foreign exchange risk premiums are correlated with interest rates. In fact,
the information from (14) indicates that the source of uncertainty is coming from the stochastic
nature of the discount factors associated with stock returns and inflation rates relative to interest
rates. Using Equations (9) and (10) and defining = , we obtain: et 1+δ )( *
1 ttet rrs −−∆ +
)()])[( *
11*
11e
te
ttet
et rrrR ++++ −−−−= αγδ (15)
The ex ante excess depreciation of a national currency beyond its interest-rate parity condition,
where is positive, is seen to be associated with relatively higher risk in stock returns and/or
inflation variations, reflected in a relatively higher equity premium and/or lower expected real
interest-rate differential. Those parameters are the main factors that cause international capital
flows. Thus, violations of UIP correspond to international capital flows.
et 1+δ
Comparing Equation (15) with existing literature, it is easy to see that the real interest-rate
differential hypothesis proposed by Korajczyk (1985) is equivalent to requiring that γ = 0, while
the equity-premium differential hypothesis suggested by Chiang (1991) is to impose the
restriction that α = 0. Of course, the UIP holds when α = γ = 0.
Next, let us consider the deviation of the international stock-return parity, defined as =
. This expression can be further decomposed as:
et 1+φ
)( *,,1etm
etm
et RRs −−∆ +
= [ . et 1+φ )]()( **
,,1 tetmt
etmt
et rRrRrs −−−−∆ +
Using the information in Equation (14), we then derive:
)()])[(1( *11
**111
et
ett
et
et
et rrrR +++++ −−−−−= αγφ (16)
14
Equation (16) indicates that the deviation of the ISP is attributable to the equity-premium
differential and real interest-rate differential. By the same token, it can be shown that:
))(1()]()[( *11
**111
et
ett
ett
et
et rrrRrR +++++ −−+−−−= αγθ (17)
where = , which denotes the ex ante value of the deviation of the
relative purchasing-power parity. By checking the right side variables of Equations (15), (16),
and (17), we observe that departures from parity conditions are all attributable to the same
factors: the equity-premium differential and the real interest-rate differential.
et 1+θ )( *
111et
et
et pps +++ ∆−∆−∆
13,14 This is
equivalent to saying that to have these parity conditions hold, the following conditions must be
satisfied: (a) expected real returns on nominal bonds are equal across markets and (b) expected
excess returns in national equity markets are equal across trading countries. The emphasis on the
real interest-rate parity to explain the departure of the three parities has been well documented
(Mishkin, 1984; Marston, 1997). However, our analysis identifies an additional factor, the
equity-premium differential, in interpreting the deviations of the three parities.
Another feature of our model is that deviations from parity conditions for the three markets
are not independent. The interdependency among them essentially is rooted in the
interdependency of financial markets, and dynamic adjustments are sensitive to differences in
relative asset returns in an integrated and united financial system. From a policy point of view,
a parametric change in interest rates made by the monetary authorities will create a gap in both
the equity-premium differential and the real interest-rate differential. These would cause
investors to reallocate their portfolios, inducing capital and trade flows and hence disturbing the
parity conditions.15
4.2. Evidence for Deviations from Parity Conditions
In this section, we present evidence for estimating the deviations from the three international
parity conditions. The estimated equation is written in the following regression form:
C. Deviation from the Relative Purchasing-Power Parity
UK 0.0023 0.297* -0.088 0.106 1.838 (1.000) (2.993) (0.195) GM 0.0026 0.131* -1.666** 0.079 1.871 (1.064) (2.901) (2.312) FR 0.0047** 0.174* -1.455*** 0.096 2.014 (1.975) (3.542) (1.803) SW 0.0002 0.232* -1.709** 0.106 1.908 (0.063) (3.642) (2.221) ____________________________________________________________ Notes: a. Numbers in parentheses are absolute value of the t-statistics. b. * and ** indicate significance at the 1% and 5% levels, respectively. c. Sample period: January 1989 – October 2001.
17
5. Conclusion
This study presents a consistent market behavior to establish three parity conditions in bond,
stock, and goods markets. Due to the existence of inflation risk and exchange rate risk, earlier
studies recognized the significance of a real interest-rate differential as a key element in
explaining deviations of interest-rate parity or purchasing-power parity. However, the real
interest-rate differential does not seem adequate to explain capital movements involving the
trading of international stocks. On the other hand, the equity-premium differential hypothesis
highlights the relative risk factor in equity markets; inflation rate uncertainty has been ignored.
In the current model, both the real interest-rate differential and the equity-premium differential
are used to explain the departures. The statistical results derived from the four European markets
relative to the United States validate our argument. The evidence further suggests that deviations
from the three international parity conditions are driven by common factors as represented by the
equity-premium differentials and real interest-rate differentials. The intriguing information
content of those differentials is that they reflect not only the relative risk across countries, but
also the relative risk as compared with fixed-income investment.
18
Appendix 1
This Appendix provides additional empirical evidence on the popular parity conditions
prevailing in the international markets. The regression models are:
A. Efficient Interest Rate Parity: +1+ts =−− )( *tt rr 0β 1β st + 1+tε
B. Efficient International Stock Parity: +1+ts =−− ++ )( *11 tt RR 0β 1β st + 1+tε
C. Efficient Purchasing Power Parity: +1+ts =∆−∆− ++ )( *11 tt pp 0β 1β st + 1+tε
D. International Fama Parity: =∆−∆ ++ )( *11 tt pp 0β + 1β +) 1+t( *
tt rr − ε
E. Real Interest Rate Parity: =+1tr 0β + 1β *1+tr + 1+tε
F. Equity Premium Parity: 1**
1101 )( +++ +−+=− ttttt rRrR εββ
G. Covered Interest parity: - tr +−+= )(10*
ttt sfr ββ tε
H. Unbiased Forward Rate Hypothesis I: −tf 1+ts = 0β + 1β )( tt sf − + 1+tε
1. Other parity conditions, including an unbiased forward-rate hypothesis, covered interest-rate parity, and real interest-rate parity will be discussed at a later point. A formal derivation of these parity conditions can be achieved by employing a consumption-based approach in the Lucas framework (Lucas, 1982; Roll and Solnik, 1979; Chiang and Trinidad, 1997, Cochrane, 2001). 2. We ignore the coupon payment (ct+1) to the bond and the dividend payment (dt+1) to the stock
by assuming ct+1 = dt+1 = 0 in order to simplify the analysis. Different tax effects are also abstracted from the calculations. We can link the current model to Lucas-Cochrane framework by setting pvjt = . Thus, = E(mtp tp t+1 xt+1), where is current asset price; mtp t+1
is the stochastic discount factor; xt+1 is the payoff at time t+1. By setting , we
have: =
11 ++ = tt px
tp ERe
t 1
1
+
t )( 1+tp .
3. An equilibrium relationship between asset returns based on a continuous time model can be found in Stulz (1981). 4. Frankel and MacArthur (1988) further decompose UIP into two parts: the covered-interest differential and the currency-risk premium. Thus, equation (11)’ becomes:
The first term on the right side of this expression is a deviation of the covered interest rate, which is considered as a country premium; the second term is the currency-risk premium; the third term is the change in the real exchange rate. Branson (1988) interprets these three components as the measure of a lack of integration of bonds, currency, and goods markets, respectively.
5. A systematic relationship between stock returns and inflation is found in Stulz’s study (1986). 6. The Basle Accord was a landmark regulatory agreement affecting international banking. The Agreement was reached on July 12, 1988. The goals were to reduce the risk of the international banking system and to minimize competitive inequality due to differences among national banking and capital regulations (Wagster, 1996). 7. Using realizations to proxy expectations could generate an error-in-the-variables problem. In
fact, the formation of expectations has long been a challenging issue in empirical estimations. It ranges from rational expectations, distributed lag expectations, adaptive expectations, regressive expectations, and random walk to expert expectations based on survey data (Frankel and Froot, 1987).
23
8. In the finance literature, expected returns are related to risk, which can be modeled by ARCH or GARCH in mean (Baillie and Bollerslev, 1990). Also, many recent studies incorporate conditional variance and covariance into various models to examine the relationship between excess returns and risk (Domowitz and Hakkio, 1985; Hodrick, 1987; Bekaert and Hodrick, 1993; Hu, 1997; De Santis and Gerard, 1998; Jiang and Chiang, 2000, Cochrane, 2001). In this paper, we do not intent to explore this type of models.
9. Our test here follows the traditional approach by focusing on examining whether the slope
coefficient differs significantly from unity. Rogoff (1996) provides a good review. Recent research pays particular attention to the stochastic properties of the dynamics of adjustments toward PPP and employs more power statistical techniques. Cheung and Lai (1993,1998), Jorion and Sweeney (1996), Lothian and Taylor (1997b), and Baum, et al. (2001) present evidence in favor of PPP.
10. Roll’s efficient estimations and other parity conditions are provided in Appendix 1. 11. Expected inflation-rate differentials can also affect the capital account through their effects on real interest rate differentials (Frankel, 1979). 12. As mentioned earlier, Frankel and MacArthur (1988) decomposed UIP into two parts: the
covered-interest differential and the currency-risk premium, while Gokey (1994) decomposed the UIP into a real interest-rate differential and an ex ante deviation from relative PPP.
[ (∆ ∆ + [=−−∆ + )( *1 ttt rrs −∆ +
ets 1 −+
etp 1 )]*
1e
tp +e
tr*1+ ]1
etr +− .
Basically, Frankel and MacArthur’s (1988) decomposition is achieved by subtracting and adding the forward premium, ( , into the UIP as we showed in note 4, while Gokey’s decomposition is obtained by subtracting and adding the expected inflation-rate differential, (∆ ∆ , into the equation.
)tt sf −
−+etp 1 )*
1e
tp +
13. The difference in long-short rate spreads can also be included on the right side of Equation (14) to capture the information of relative liquidity risk as implied by the expectations hypothesis of the term-structure of interest rates.
14. Using Equations (15)-(17), we obtain the following two equations as (11)’ and (12)’:
et
et rr *
11 ++ − = - and et 1+θ e
t 1+δ
= - )()( **11 t
ett
et rRrR −−− ++
et 1+δ e
t 1+φ
15. A precise process and speed of adjustment to restore a new equilibrium can be very complicated and so cannot be answered without having a complete specification of the model, which is beyond the scope of current study.
24
16. Cumby (1990) tests whether real stock returns from four countries are consistent with consumption-based models of international asset pricing. The hypothesis was rejected by including a sample that began in 1974. However, the null cannot be rejected when only the 1980s are considered.
17. Estimates of the unbiasedness hypothesis are based on the sample period from 1989.1 to
1998.12 due to unavailability of FR, GM, and SW forward markets and the switch to the Euro starting in January 1999.
25
References
Abuaf, N. and Jorion, P. (1990) Purchasing power parity in the long run. Journal of Finance, 45, 157-174.
Baillie, R.T. and Bollerslev, T. (1990) A multivariate generalized ARCH approach to modeling risk premia in forward foreign exchange rate markets. Journal of International Money and Finance, 9, 309-324.
Baum, C. F., Barkoulas, J.T., and Caglayan, M. (2001) Nonlinear adjustment to purchasing power parity in the post-Bretton Woods era. Journal of International Money and Finance, 20, 379-399.
Bekaert, G. and Hodrick, R.J. (1993) On biases in the measurement of foreign exchange risk premiums. Journal of International Money and Finance, 12, 115-138.
Branson, W.H. (1988). Comments on political vs. currency premia in international real interest differentials: a study of forward rates for 24 countries, by Frankel, J.A. and MacArthur, A.T., European Economic Review, 32, 1083-1114.
Campbell, J.Y. (1987) Stock returns and the term structure. Journal of Financial Economics, 18, 373-399.
Cheung, Y. W, and Lai, K.S. (1993) Long-run purchasing power parity during the recent float. Journal of International Economics, 34, 181-192.
Cheung, Y. W, and Lai, K.S. (1998) Parity reversion in real exchange rates during the post-Bretton Woods period. Journal of International Money and Finance, 17, 597-614.
Chiang, T.C. (1991) International asset pricing and equity market risk. Journal of International Money and Finance, 10, 349-364.
Chiang, T.C. and Trinidad, J. (1997) Risk and international parity conditions: a synthesis from consumption-based models. International Economic Journal, 11, 73-101.
Cochrane, J.H. (2001) Asset Pricing. Princeton and Oxford: Princeton University Press.
Cornell, B. (1989) The impact of data errors on measurement of the foreign exchange risk premium. Journal of International Money and Finance, 8, 147-157.
Cumby, R. (1990) Consumption risk and international equity returns: some empirical evidence. Journal of International Money and Finance, 9, 182-192.
De Santis and Gerard, B. (1998) How big is the premium for currency risk? Journal of Financial Economics, 49 375-412.
26
Domowitz, I. and Hakkio, C. S. (1985) Conditional variance and the risk premium in the Foreign exchange market. Journal of International Economics, 19, 47-66.
Engel, C. (1996) The forward discount anomaly and the risk premium: a survey of recent evidence. Journal of Empirical Finance, 3, 123-192.
Fama, E.F. (1975) Short-term interest rates as predictors of inflation. American Economic Review, 65, 269-282.
Fama, E.F. (1984) Forward and spot exchange rates. Journal of Monetary Economics, 14, 319-338.
Fisher, E. and Park, J. (1991) Testing purchasing power parity under the null hypothesis of cointegration. Economic Journal, 101, 1476-1484.
Fleissig, A.R. and Strauss, J. (2000) Panel unit root tests of purchasing power parity for price indices. Journal of International Money and Finance, 19, 489-7506.
Frankel, J.A. (1979) On the mark: a theory of floating exchange rates based on real interest differential. American Economic Review, 69, 610-622.
Frankel, J.A. and Froot, K.A. (1987) Using survey data to test standard propositions regarding exchange rate expectations. American Economic Review, 77, 133-153.
Frankel, J.A. and MacArthur, A.T. (1988) Political vs. currency premia in international real interest differentials: A study of forward rates for 24 countries. European Economic Review, 32, 1083-1114.
Frankel, J. A. and Schmukler, A. L. (2000) Country funds and asymmetric information. International Journal of Finance and economics, 5, 177-195.
Fratianni, M. and Wakeman, L.M. (1982) The law of one price in the Eurocurrency market. Journal of International Money and Finance, 3, 307-323.
Frenkel, J.A. (1981) The collapse of purchasing power parities during the 1970s. European Economic Review, 16, 145-165.
Froot, K. A. and Thaler, R.H. (1990) Anomalies: foreign exchange. The Journal of Economic Perspective, 4, 179-192.
Giovannini, A. and Jorion P. (1987) Interest rates and risk premia in the stock market and in the foreign exchange market. Journal of International Money and Finance, 6, 107-124.
Gokey, T.C. (1994) What explains the risk premium in foreign exchange returns. Journal of International Money and Finance, 13, 729-738.
27
Hansen, L.P. and Hodrick, R.J. (1980) Forward exchange rates as optimal predictors of future spot rates: an econometric analysis. Journal of Political Economy, 88, 829-853.
Hodrick, R.J. (1987) The Empirical Evidence On The Efficiency Of Forward And Futures Foreign Exchange Markets. New York: Harwood Academic.
Hu, X. (1997) Macroeonomic uncertainty and the risk premium in the foreign exchange market. International Journal of Finance and Finance, 16, 699-718.
Huang, R.D. (1990) Risk and parity in purchasing power. Journal of Money, Credit and Banking, 22, 338-356.
Jiang, C. and Chiang, T.C. (2000) Do foreign exchange risk premiums relate to the volatility in the foreign exchange and equity markets? 10, 95-104.
Jorian, P. and Sweeney, R.J. (1996) Mean reversion in real exchange rates, International Journal of Finance and Economics, 15, 535-550.
Korajczyk, R.A. (1985) The pricing of forward contracts for foreign exchange. Journal of Political Economy, 93, 346-368.
Korajczyk, R. A. and Viallet, C. J. (1992) Equity risk premia and the pricing of foreign exchange risk. Journal of International Economics, 33, 199-220.
Krugman, P. (1978) Purchasing power parity and exchange rate, another look at the evidence. Journal of International Economics, 8, 397-407.
Levine, R. (1989) The pricing of forward exchange rates. Journal of International Money and Finance, 8, 163-180.
Lothian,, J.R. and Taylor, M.P. (1997a) The recent float from the perspective of the past two centuries, Journal of Political Economy, 104, 488-509.
Lothian, J.R. and Taylor, M.P. (1997b) Real exchange rate behavior. Journal of International Money and Finance, 16, 945-954.
Lucas, R.E. Jr. (1982) Interest rates and currency prices in a two-country world, Journal of Monetary Economics, 10, 335-360.
Mark, N.C. (1988) Time-varying betas and risk premia in the pricing of forward foreign exchange contracts. Journal of Financial Economics, 22, 335-354.
Marston, R.C. (1997) Tests of three parity conditions: distinguishing risk premia and systematic forecast errors. Journal of International Money and Finance, 16, 285-303.
Mishkin, F.S. (1983) A rational expectations approach to macroeconometrics: testing policy ineffectiveness and efficient markets models. Chicago: University of Chicago Press.
28
29
Mishkin, F.S. (1984) Are real interest rates equal across countries: an empirical investigation of international parity conditions. Journal of Finance, 39, 1345-1357.
Newey, W. K. and West, K. D. (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55, 703-708.
Patel, J. (1990) Purchasing power parity as a long run relation, Journal of Applied Econometrics, 5, 367-379.
Rogoff, K. (1996). The purchasing power parity puzzle. Journal of Economic Literature, 34, 647-668.
Roll, R. (1979) Violations of purchasing power parity and their implications for efficient international commodity market. In M. Sarnat and G. P. Szego, eds. International Finance and Trade. Cambridge, MA: Ballinger.
Roll, R. and Solnik, B. (1979) On some parity conditions encountered frequently in international economics. Journal of Macroeconomics, 1, 267-283.
Roll, R. (1992) Industrial structure and the comparative behavior of international stock market indexes. Journal of Finance, 47, 3-41.
Solnik, B. (1978) International parity conditions and exchange risk. Journal of Banking and Finance, 2, 281-293.
Solnik, B. (1982) An empirical investigation of the determinants of national interest rate differences. Journal of International Money and Finance, 3, 333-339.
Stulz, R, M. (1981) A model of international asset pricing. Journal of Financial Economics, 9, 383-406.
Stulz, R, M. (1986) Asset pricing and expected inflation. Journal of Finance, 41, 209-223.
Taylor, M. (1988) An empirical examination of long run purchasing power parity using cointegration techniques. Applied Economics, 20, 1369-1381.
Wagster., J.D. (1996) Impact of the 1988 Basle Accord on international banks, Journal of Finance, 51, 1321-1346.
White, H. (1980) A heteroscedasticity-consistent covariance matrix estimator and a direct test for heteroscedasticity. Econometrica, 48, 55-68.
Wu, Y. and Zhang, H. (1997) Forward premiums as unbiased predictors of future currency depreciation: a non-parametric analysis. Journal of International Money and Finance, 16, 609-623.